[SalesForce] SOQL to find whether user’s role is part of the public group

that will do the trick. what you basically have to do is to traverse through the nested groups by self referencing the getGroupsForIds() method

// return list of all groups the user belongs to via direct or indirect membership
public Group[] getGroupsForUser(Id userId){

    Set<Id> groupIds = getGroupsForIds(new Set<Id>{userId});
 return [
   select Id
        , Name
     from Group
    where Id IN: groupIds];

}

// return all ids the user belongs to via direct or indirect membership
public Set<Id> getGroupsForIds(Set<Id> userOrGroupIds){

    Set<Id> output = new Set<Id>();

    Set<Id> nestedGroupIds = new Set<Id>();

    // only query actual groups and not roles and queues
    list<GroupMember> records = [
        select id
             , GroupId
             , UserOrGroupId
          from GroupMember
         where UserOrGroupId =: userOrGroupIds
        and UserOrGroupId != null
           and Group.Type = 'Regular'];

    for (GroupMember record:records)
    {
        // found a group, remember for traversal
        if (!(record.UserOrGroupId + '').startsWith('005'))
        {
            nestedGroupIds.add(record.UserOrGroupId);   
        }
        else
        {
            output.add(record.GroupId);
        }
    }

    // call self to get nested groups we found
    if (nestedGroupIds.size() > 0)
    {
        output.addAll(getGroupsForIds(nestedGroupIds));
    }

    return output;
}

My experimentation seems to indicate that for a UserRole you add to a Group, Salesforce adds a Group with Type = 'Role' and RelatedId = <UserRole.Id>. You may need to separately process RoleAndSubordinates and/or RoleAndSubordinatesInternal, depending on your requirements.

Regardless, you should be able to query for the Id of the UserRole by iterating through the GroupMember records and filtering by Type. As there are many Type values to inspect, if you want to properly handle all of them you may need to do more legwork, as you shouldn't trust my back-of-the-napkin implementation. But it should get you started, hopefully.

Set<Id> roles = new Set<Id>();
Set<Id> rolesWithSubordinate = new Set<Id>();

Set<Id> groupIds = new Set<Id>();
for (GroupMember member : myGroup)
    groupIds.add(member.UserOrGroupId);

for (Group record : [
    SELECT Type, RelatedId FROM Group WHERE Id IN :groupIds
]){
    switch on record.Type
    {
        when 'Role'
        {
            roles.add(record.RelatedId);
        }
        when 'RoleAndSubordinates'
        {
            rolesWithSubordinate.add(record.RelatedId);
        }
    }
}
// query users with the above roles here